Nonconvex reformulations via low-rank factorization for stochastic convex semidefinite optimization problem have attracted arising attention due to their empirical efficiency and scalability. Compared with the original convex formulations, the nonconvex ones typically involve much fewer variables, allowing them to scale to scenarios with millions of variables. However, it opens a new challenge that under what conditions the nonconvex stochastic algorithms may find the population minimizer within the optimal statistical precision despite their empirical success in applications. In this paper, we provide an answer that the stochastic gradient descent (SGD) method can be adapted to solve the nonconvex reformulation of the original convex problem, with a global linear convergence when using a fixed step size, i.e., converging exponentially fast to the population minimizer within an optimal statistical precision in the restricted strongly convex case. If a diminishing step size is adopted, the bad effect caused by the variance of gradients on the optimization error can be eliminated but the rate is dropped to be sublinear. The core of our treatment relies on a novel second-order descent lemma, which is more general than the existing best result in the literature and improves the analysis on both online and batch algorithms. The developed theoretical results and effectiveness of the suggested SGD are also verified by a series of experiments.

We use techniques from both real and complex algebraic geometry to study$K$-theoretic and related invariants of the algebra$C$($X$) of continuous complex-valued functions on a compact Hausdorff topological space$X$. For example, we prove a parameterized version of a theorem by Joseph Gubeladze; we show that if$M$is a countable, abelian, cancellative, torsion-free, semi-normal monoid, and$X$is contractible, then every finitely generated projective module over$C$($X$)[$M$] is free. The particular case $$ M = \mathbb{N}_0^n $$ gives a parameterized version of the celebrated theorem proved independently by Daniel Quillen and Andrei Suslin that finitely generated projective modules over a polynomial ring over a field are free. The conjecture of Jonathan Rosenberg which predicts the homotopy invariance of the negative algebraic$K$-theory of$C$($X$) follows from the particular case $$ M = {\mathbb{Z}^n} $$ . We also give algebraic conditions for a functor from commutative algebras to abelian groups to be homotopy invariant on$C$^{*}-algebras, and for a homology theory of commutative algebras to vanish on$C$^{*}-algebras. These criteria have numerous applications. For example, the vanishing criterion applied to nil$K$-theory implies that commutative$C$^{*}-algebras are$K$-regular. As another application, we show that the familiar formulas of Hochschild–Kostant–Rosenberg and Loday–Quillen for the algebraic Hochschild and cyclic homology of the coordinate ring of a smooth algebraic variety remain valid for the algebraic Hochschild and cyclic homology of$C$($X$). Applications to the conjectures of Beĭlinson-Soulé and Farrell–Jones are also given.

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Surface flow phenomena, such as rain water flowing down a tree trunk and progressive water front in a shower room, are common in real life. However, compared with the 3D spatial fluid flow, these surface flow problems have been much less studied in the graphics community. To tackle this research gap, we present an efficient, robust and high-fidelity simulation approach based on the shallow-water equations. Specifically, the standard shallow-water flow model is extended to general triangle meshes with a feature-based bottom friction model, and a series of coherent mathematical formulations are derived to represent the full range of physical effects that are important for real-world surface flow phenomena. In addition, by achieving compatibility with existing 3D fluid simulators and by supporting physically realistic interactions with multiple fluids and solid surfaces, the new model is flexible and readily extensible for coupled phenomena. A wide range of simulation examples are presented to demonstrate the performance of the new approach.

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